Bootstrapping Aalen-Johansen processes for competing risks: Handicaps, solutions, and limitations

Abstract

Statistical inference in competing risks models is often based on the famous Aalen-Johansen estimator. Since the corresponding limit process lacks independent increments, it is typically applied together with Lin's (1997) resampling technique involving standard normal multipliers. Recently, it has been seen that this approach can be interpreted as a wild bootstrap technique and that other multipliers, as e.g. centered Poissons, may lead to better finite sample performances, see Beyersmann et al. (2013). Since the latter is closely related to Efron's classical bootstrap, the question arises whether this or more general weighted bootstrap versions of Aalen-Johansen processes lead to valid results. Here we analyze their asymptotic behaviour and it turns out that such weighted bootstrap versions in general possess the wrong covariance structure in the limit. However, we explain that the weighted bootstrap can nevertheless be applied for specific null hypotheses of interest and also discuss its limitations for statistical inference. To this end, we introduce different consistent weighted bootstrap tests for the null hypothesis of stochastically ordered cumulative incidence functions and compare their finite sample performance in a simulation study.

Non-strange weird resampling for complex survival data

“…and Z 2n;i := X n;1;i 1(i ≤ n) + X n;2;i−n 1(i > n). That is, we obtain a linear weighted representation as in Dobler and Pauly (2014). Now replacing the i.i.d.…”

“…The variance n var( F 1 (t)) in the DDMB resampling version γ n is similar to the wild bootstrap variance estimator of Dobler and Pauly (2014), where we now use the same DDMB weights as in W D n . Again following Lin (1997) and Beyersmann et al (2013) we call the bands resulting from g 1 and g 2 equal precision and Hall-Wellner bands, respectively.…”

A discontinuity adjustment for subdistribution function confidence bands applied to right-censored competing risks data

“…These methods are compared in terms of the simulated coverage probabilities of the confidence bands described in Remark 4.4. We consider a simulation set-up motivated by Dobler and Pauly (2014), i.e. we chose the cause-specific hazard rates α 1 ptq " expp´tq and α 2 ptq " 1´expp´tq which yield the cumulative function of the first risk F 1 ptq " 0.5p1´expp´2tqq.…”

Confidence interval estimation for the changepoint of treatment stratification in the presence of a qualitative covariate‐treatment interaction

“…Now, to obtain a wild bootstrap version of this, we replace all martingale increments d M in the score function U n by ξ d N , with independent multipliers ξ for the counting processes of different individuals. Moreover, we replace in I n the counting processes d N with ξ 2 d N ; see Dobler and Pauly for a similar approach in the context of nonparametric cumulative incidence functions. It was shown that using these squared multipliers corresponds to using the optional variation process of the bootstrapped processes when considered as martingales in time .…”